Set \(\mathcal {W}:=\operatorname{Supp}_R(M)\)  and \(\operatorname{\mathcal {Z}}^{\prime }:=\operatorname{\mathcal {Z}}\cap \mathcal {W}\) . Clearly, \(\operatorname{\mathcal {Z}}^{\prime }\)  is specialization-closed. First we prove:
Claim I. \(\operatorname{H}^i_{\operatorname{\mathcal {Z}}}(M\otimes _RN)\cong \operatorname{H}^i_{\operatorname{\mathcal {Z}}^{\prime }}(M\otimes _RN)\)  for all \(i\ge 0\) .
Proof of Claim. Set \(L=M\otimes _RN\)  and let \(0\rightarrow L\rightarrow \operatorname{E}^0(L)\rightarrow \operatorname{E}^1(L)\rightarrow \cdots \)  be the minimal injective resolution of \(X\) . Note that \(\operatorname{Supp}_R(\operatorname{E}^i(L))\subseteq \operatorname{Supp}_R(L)\subseteq \mathcal {W}\)  for all \(i\ge 0\) . Therefore, \(\Gamma _{\mathcal {W}}(L)=L\)  and also \(\Gamma _{\mathcal {W}}(\operatorname{E}^i(L))=\operatorname{E}^i(L)\)  for all \(i\ge 0\) .
It follows from Theorem REF (iv) that \(\Gamma _{\operatorname{\mathcal {Z}}^{\prime }}(L)=\Gamma _{\operatorname{\mathcal {Z}}}(\Gamma _{\mathcal {W}}(L))=\Gamma _{\operatorname{\mathcal {Z}}}(L)\) . Similarly,
\(\Gamma _{\operatorname{\mathcal {Z}}^{\prime }}(\operatorname{E}^i(L))=\Gamma _{\operatorname{\mathcal {Z}}}(\Gamma _{\mathcal {W}}(\operatorname{E}^i(L)))=\Gamma _{\operatorname{\mathcal {Z}}}(\operatorname{E}^i(L))\)  for all \(i\ge 0\)  which implies that \(\operatorname{H}^i_{\operatorname{\mathcal {Z}}}(L)\cong \operatorname{H}^i_{\operatorname{\mathcal {Z}}^{\prime }}(L)\)  for all \(i\ge 0\) . Thus, the proof of the claim is completed.
By Claim I, without loss of generality, by replacing \(\operatorname{\mathcal {Z}}\)  with \(\operatorname{\mathcal {Z}}^{\prime }\) , we may assume that \(\operatorname{\mathcal {Z}}\subseteq \operatorname{Supp}_R(M)\) . Consider the following exact sequence
\(0\rightarrow M\rightarrow X\rightarrow G\rightarrow 0\qquad \mathrm {(\ref {t12}.1)}\) 
where \(\operatorname{pd}_R(X)<\infty \)  and \(-dim_R(G)=0\)  (see [1]}).
By [2]}, the exact sequence (REF .1) induces a
doubly infinite long exact sequence
\(\cdots \rightarrow \widehat{\operatorname{Tor}}_{j+1}^R(G,N)\rightarrow \widehat{\operatorname{Tor}}_{j}^R(M,N)\rightarrow \widehat{\operatorname{Tor}}_{j}^R(X,N)\rightarrow \widehat{\operatorname{Tor}}_{j}^R(G,N)\rightarrow \cdots ,\) 
of stable homology modules. Also by Theorem REF (i), \(\widehat{\operatorname{Tor}}_{j}^R(X,N)=0\)  for all \(j\in \mathbb {Z}\) . Hence we get the following isomorphism
\(\widehat{\operatorname{Tor}}_{j+1}^R(G,N)\cong \widehat{\operatorname{Tor}}_{j}^R(M,N) \text{ for all } j\in \mathbb {Z}.\qquad \mathrm {(\ref {t12}.2)}\) 
As \(G\)  is totally reflexive, by Theorem REF (iv), (REF .2) and assumption (ii), we have
\(\operatorname{Supp}_R(\operatorname{Tor}_i^R(G,N))=\operatorname{Supp}_R(\widehat{\operatorname{Tor}}_i^R(G,N))=\operatorname{Supp}_R(\widehat{\operatorname{Tor}}_{i-1}^R(M,N))\subseteq \operatorname{\mathcal {Z}}\text{ for all } i>0.\qquad \mathrm {(\ref {t12}.3)}\) 
Applying the functor \(-\otimes _RN\)  to the exact sequence (REF .1), we get the following exact sequence
\(\cdots \rightarrow \operatorname{Tor}_i^R(M,N)\rightarrow \operatorname{Tor}_i^R(X,N)\rightarrow \operatorname{Tor}_i^R(G,N)\rightarrow \cdots .\qquad \mathrm {(\ref {t12}.4)}\) 
In view of assumption (ii), (REF .3) and (REF .4) we have
\(\operatorname{Supp}_R(\operatorname{Tor}_i^R(X,N))\subseteq \operatorname{Supp}_R(\operatorname{Tor}_i^R(M,N))\cup \operatorname{Supp}_R(\operatorname{Tor}_i^R(G,N))\subseteq \operatorname{\mathcal {Z}}\text{ for all } i>0.\qquad \mathrm {(\ref {t12}.5)}\) 
On the other hand, by the exact sequence (REF .1) and Auslander–Bridger formula, we see that \(\operatorname{depth}_{R_{\mathfrak {p}}}(M_\mathfrak {p})=\operatorname{depth}_{R_{\mathfrak {p}}}(X_\mathfrak {p})\)  for all \(\mathfrak {p}\in \operatorname{\mathcal {Z}}\subseteq \operatorname{Supp}_R(M)\) . Therefore, by assumption (i)
\(\operatorname{depth}_{R_{\mathfrak {p}}}(X_\mathfrak {p})+\operatorname{depth}_{R_{\mathfrak {p}}}(N_\mathfrak {p})\ge \operatorname{depth}R_{\mathfrak {p}}+ n \quad \forall \mathfrak {p}\in \operatorname{\mathcal {Z}}.\qquad \mathrm {(\ref {t12}.6)}\) 
Next we claim the following.
Claim II. \(\operatorname{Tor}_i^R(X,N)=0\)  for all \(i>0\) .
Proof of claim. Set \(t=\sup \lbrace j\mid \operatorname{Tor}_{j}^R(X,N))\ne 0\rbrace \) . Assume contrarily that \(t>0\)  and that \(\mathfrak {p}\in \operatorname{Ass}_R(\operatorname{Tor}_t^R(X,N))\) . By (REF .5), \(\mathfrak {p}\subseteq \operatorname{\mathcal {Z}}\) . As \(\operatorname{depth}_{R_\mathfrak {p}}(\operatorname{Tor}_t^R(X,N)_\mathfrak {p})=0\) , by [3]} we have the equality
\(\operatorname{depth}_{R_\mathfrak {p}}(X_\mathfrak {p})+\operatorname{depth}_{R_\mathfrak {p}}(N_\mathfrak {p})=\operatorname{depth}R_\mathfrak {p}-t,\) 
which is a contradiction by (REF .6). Therefore \(t=0\)  and the proof of the claim is completed.
By Claim II, the long exact sequence (REF .4) induces the following exact sequence
\(0\rightarrow \operatorname{Tor}_1^R(G,N)\rightarrow M\otimes _RN\rightarrow X\otimes _RN\rightarrow G\otimes _RN\rightarrow 0.\qquad \mathrm {(\ref {t12}.7)}\) 
Applying the functor \(\Gamma _{\operatorname{\mathcal {Z}}}(-)\)  to the exact sequence (REF .7), we obtain the following injection:
\(\operatorname{H}^0_{\operatorname{\mathcal {Z}}}(\operatorname{Tor}_1^R(G,N))\hookrightarrow \operatorname{H}^0_{\operatorname{\mathcal {Z}}}(M\otimes _RN).\qquad \mathrm {(\ref {t12}.8)}\) 
Note by (REF .3) that \(\operatorname{Tor}_1^R(G,N)\)  is torsion with respect to \(\operatorname{\mathcal {Z}}\) . Hence, by (REF .2), Theorem REF (iv) and (REF .8) we get the following injection:
\(\widehat{\operatorname{Tor}}_0^R(M,N)\cong \widehat{\operatorname{Tor}}_1^R(G,N)\cong \operatorname{Tor}_{1}^R(G,N)\cong \operatorname{H}^0_{\operatorname{\mathcal {Z}}}(\operatorname{Tor}_1^R(G,N))\hookrightarrow \operatorname{H}^0_{\operatorname{\mathcal {Z}}}(M\otimes _RN).\) 
Therefore, from now on we may assume that \(n>0\) . By Theorem REF  and Claim II,
\(\operatorname{depth}_{R_\mathfrak {p}}(X_\mathfrak {p})+\operatorname{depth}_{R_\mathfrak {p}}(N_\mathfrak {p})=\operatorname{depth}R_{\mathfrak {p}}+\operatorname{depth}_{R_\mathfrak {p}}((X\otimes _RN)_\mathfrak {p})\quad \forall \mathfrak {p}\in \operatorname{Supp}_R(X\otimes _RN).\qquad \mathrm {(\ref {t12}.9)}\) 
In view of (REF .6) and (REF .9), we have
\(\operatorname{depth}_{R_\mathfrak {p}}((X\otimes _RN)_\mathfrak {p})\ge n\)  for all \(\mathfrak {p}\in \operatorname{\mathcal {Z}}\) . By Proposition  REF
\(\operatorname{H}^i_{\operatorname{\mathcal {Z}}}(X\otimes _RN)=0 \text{ for all } 0\le i\le n-1.\qquad \mathrm {(\ref {t12}.10)}\) 
Consider the following two exact sequences, induce from (REF .7):
\(0\rightarrow \operatorname{Tor}_1^R(G,N)\rightarrow M\otimes _RN\rightarrow Y\rightarrow 0\ \ \text{ and }\ \ 0\rightarrow Y\rightarrow X\otimes _RN\rightarrow G\otimes _RN\rightarrow 0.\qquad \mathrm {(\ref {t12}.11)}\) 
Applying the functor \(\Gamma _{\operatorname{\mathcal {Z}}}(-)\)  to the above exact sequences and using Theorem REF (ii), (iii), (REF .10) and (REF .2), one can easily obtain the following isomorphisms:
\(\operatorname{H}^0_{\operatorname{\mathcal {Z}}}(M\otimes _RN)\cong \operatorname{H}^0_{\operatorname{\mathcal {Z}}}(\operatorname{Tor}_1^R(G,N))\cong \operatorname{Tor}_1^R(G,N)\cong \widehat{\operatorname{Tor}}_1^R(G,N)\cong \widehat{\operatorname{Tor}}_0^R(M,N),\qquad \mathrm {(\ref {t12}.12)}\) 
\(\operatorname{H}^i_{\operatorname{\mathcal {Z}}}(M\otimes _RN)\cong \operatorname{H}^i_{\operatorname{\mathcal {Z}}}(Y)\cong \operatorname{H}^{i-1}_{\operatorname{\mathcal {Z}}}(G\otimes _RN) \text{ for all } 0<i<n.\qquad \mathrm {(\ref {t12}.13)}\) 
Also, we have the following injection
\(\operatorname{H}^{n-1}_{\operatorname{\mathcal {Z}}}(G\otimes _RN)\hookrightarrow \operatorname{H}^n_{\operatorname{\mathcal {Z}}}(Y)\cong \operatorname{H}^n_{\operatorname{\mathcal {Z}}}(M\otimes _RN).\qquad \mathrm {(\ref {t12}.14)}\) 
Note by (REF .2) and assumption (ii) that \(\operatorname{Supp}_R(\widehat{\operatorname{Tor}}^R_i(G,N))\subseteq \operatorname{\mathcal {Z}}\)  for all \(i\in \mathbb {Z}\) . As \(G\)  is totally reflexive, by Lemma REF  we have \(\operatorname{Supp}_R(\operatorname{Ext}^i_R(\operatorname{Tr}G,N))\subseteq \operatorname{\mathcal {Z}}\)  for all \(i>0\) . As \(M\)  has finite Gorenstein dimension, by assumption (i) and Auslander–Bridger formula, we have \(\operatorname{depth}_{R_\mathfrak {p}}(N_\mathfrak {p})\ge -dim_{R_\mathfrak {p}}(M_\mathfrak {p})+n\ge n\)  for all \(\mathfrak {p}\in \operatorname{\mathcal {Z}}\subseteq \operatorname{Supp}_R(M)\) . Therefore, by Proposition REF  we conclude that \(\operatorname{\textnormal {grade}}_R(\operatorname{\mathcal {Z}},N)\ge n\) . Since \(G\)  is totally reflexive, so is \(\operatorname{Tr}G\)  by [4]}.
Hence, by using Theorem REF  and Lemma REF  and noting that \(G^*\approx \Omega ^2\operatorname{Tr}G\) , we get the following isomorphisms:
\(\begin{array}{rl}\operatorname{H}^{j-1}_{\operatorname{\mathcal {Z}}}(G\otimes _RN)&\cong \operatorname{Ext}^j_R(\operatorname{Tr}G,N)\\&\cong \widehat{\operatorname{Ext}}^j_R(\operatorname{Tr}G,N)\\&\cong \widehat{\operatorname{Ext}}^{j-2}_R(G^*,N)\\&\cong \widehat{\operatorname{Tor}}_{-j+1}^R(G,N),\end{array}\qquad \mathrm {(\ref {t12}.15)}\) 
for all \(1\le j\le n\) . Now the first assertion is clear by (REF .12), (REF .13), (REF .15) and (REF .2).
The second assertion follows from (REF .14), (REF .15) and (REF .2).
